A novel approach towards bipolar complex fuzzy sets and their applications in generalized similarity measures

Abstract

The notions of the bipolar complex fuzzy set (BCFS) and complex bipolar fuzzy set (CBFS) have been already given, but these notions of BCFS and CBFS have the problem that they contradict the basic definition of the complex numbers which we discussed in this article, and then we defined the new definition of BCFS. Our defined notion of BCFS is more closed to bipolarity as compared with already existing BCFS and CBFS, and more accurate. BCFS is the fusion of bipolar fuzzy set (BFS) which a decision analyst needs to describe the positive and negative aspects of an object and complex fuzzy set (CFS) which a decision analyst needs to handle two-dimensional (two variables) information. When there is information of two variables with positive and negative aspects then a decision analyst needs BCFS to handle this information. In this article, we also interpreted some basic operations on BCFS like a complement, intersection, and union and explained them with the help of examples. Additionally, we defined the concept of type-1 partially BCFS and type-2 partially BCFS. Further, we interpreted some generalized trigonometric similarity measures such as generalized cosine similarity measure, generalized tangent similarity measure, generalized cotangent similarity measure, and generalized hybrid trigonometric similarity measure for BCFS. The weighted generalized trigonometric similarity measures are also presented in this article. After that, we applied these similarity measures (SMs) in two real-life applications (pattern recognition and medical diagnosis) to show the benefits and advantages of our proposed SMs. Finally, we did a comparison of our demonstrated SMs with some existing SMs to show the superiority, usefulness, and effectiveness of our proposed SMs.